A230892 Terms of A230891 written in base 10: the binary expansions of a(n) and a(n+1) taken together can be rearranged to form a palindrome.

0, 3, 1, 2, 4, 7, 8, 5, 6, 9, 10, 12, 15, 16, 11, 13, 14, 17, 18, 20, 23, 24, 27, 29, 30, 32, 19, 21, 22, 25, 26, 28, 31, 33, 34, 36, 39, 40, 43, 45, 46, 48, 51, 53, 54, 57, 58, 60, 63, 64, 35, 37, 38, 41, 42, 44, 47, 49, 50, 52, 55, 56, 59, 61, 62, 65, 66, 68, 71

Offset

0,2

Comments

See A230891 for precise definition.

Just as for A228407, we can ask: does every number appear? The answer is yes - see the Comments in A228407.

The difference d(n)=a(n)-n increases from d(3*2^(k-2)+2) = 1-2^(k-2) to d(3*2^(k-1)+1) = 1-2^(k-1), going through 0 at n=2^k+1 and n=2^k+2, cf. examples. - M. F. Hasler, Nov 12 2013

From Robert G. Wilson v, Nov 15 2013: (Start)

Beginning with k=3, each "grouping" of ever increasing terms, begins at 2^k + 3 and runs up to 2^(k+2) and includes 3*2^(k-1) terms.

Indices of powers of 2 occur at: 2, 3, 4, 6, 13, 25, 49, 97, 193, 385, 769, 1537, ..., which, except for 2, 3 & 6, is A181565: 3*2^n + 1.

When the index equals the term: 0, 4, 9, 10, 17, 18, 33, 34, 65, 66, 129, 130, 257, 258, 513, 514, 1025, 1026, 2049, 2050, ..., .

Parity of a(n) beginning at n=0: 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, ..., . (End)

Links

Chai Wah Wu, Table of n, a(n) for n = 0..30000 (first 2051 terms from Robert G. Wilson v)

Robert G. Wilson v, Graph of the first 1025 terms

Example

From M. F. Hasler, Nov 12 2013: (Start)
Let d(n)=a(n)-n, i.e., a(n)=n+d(n). Then we have, after initial values d(0..8)=(0, 2, -1, -1, 0, 2, 2, -2, -2), the
following pattern: d(9) = d(10) = 0, ..., d(13) = 3,
d(14) = -3, ..., d(17) = d(18) = 0, ..., d(25) = 7;
d(26) = -7, ..., d(33) = d(34) = 0, ..., d(49) = 15,
d(50) = -15, ..., d(65) = d(66) = 0, ..., d(97) = 31,
d(98) = -31, ..., d(129) = d(130) = 0, ..., d(193) = 63,
d(194) = -63,..., d(257) = d(258) = 0, ... (End)

Prog

(PARI) {u=0; a=0; La=1; ha=0/*hack*/; for(n=1, 99, u+=1<<a; print1(a", "); L=1; for(k=1, 9e9, bittest(u, k)&&next; while(k>=2^L, L++); bittest(ha+h=hammingweight(k), 0)&&!bittest(La+L, 0)&&next; !a&&k<3&&next; a=k; ha=h; La=L; break))} \\ M. F. Hasler, Nov 11 2013
(Python)
from collections import Counter
A230892_list, l, s, b = [0, 3], Counter('11'), 1, {3}
for _ in range(30001):
    i = s
    while True:
        if i not in b:
            li, o = Counter(bin(i)[2:]), 0
            for d in (l+li).values():
                if d % 2:
                    if o > 0:
                        break
                    o += 1
            else:
                A230892_list.append(i)
                l = li
                b.add(i)
                while s in b:
                    b.remove(s)
                    s += 1
                break
        i += 1 # Chai Wah Wu, Jun 19 2016

Crossrefs

Cf. A230891, A228407.

Sequence in context: A233904 A292576 A083275 * A325671 A138382 A123375

Adjacent sequences: A230889 A230890 A230891 * A230893 A230894 A230895

Keyword

nonn,base

Author

N. J. A. Sloane, Nov 11 2013

Status

approved